If there were only two wires, the resistance in the both wires would be added to the element resistance (as far as the bridg is concerend). The change in bridge voltage as a function of temperature change would be reduced.

By adding a third wire, the entire lower half of the bridge can go into the soldering iron housing. Each side has a wire going back to the rest of the circuitry, so conductor resistance cancels out. Delta V per Delta degree is thereby maximised.